Approach to the Singularity in Spatially Inhomogeneous Cosmologies

TL;DR

This paper combines analytic and numerical techniques to analyze the behavior of inhomogeneous cosmologies near singularities, supporting the dominance of local Kasner or Mixmaster dynamics as claimed by Belinskii, Khalatnikov, and Lifshitz.

Contribution

It introduces the Method of Consistent Potentials to verify the asymptotic velocity term dominance in inhomogeneous cosmologies, providing a rigorous framework for understanding singularity approaches.

Findings

Supports the BKL claim of local Kasner or Mixmaster behavior near singularities

Establishes the validity of AVTD in inhomogeneous models using the Method of Consistent Potentials

Shows how exponential terms influence the approach to singularity through dynamical potentials

Abstract

A combination of analytic and numerical methods has yielded a clear understanding of the approach to the singularity in spatially inhomogeneous cosmologies. Strong support is found for the longstanding claim by Belinskii, Khalatnikov, and Lifshitz that the collapse is dominated by local Kasner or Mixmaster behavior. The Method of Consistent Potentials is used to establish the consistency of asymptotic velocity term dominance (AVTD) (local Kasner behavior) in that no terms in Einstein's equations will grow exponentially when the VTD solution, obtained by neglecting all terms containing spatial derivatives, is substituted into the full equations. When the VTD solution is inconsistent, the exponential terms act dynamically as potentials either to drive the system into a consistent AVTD regime or to maintain an oscillatory approach to the singularity.